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User blog:KthulhuHimself/Yet another attempt at making an interesting function.
Well, I'm back, and I have a new idea for a function to show everyone. I am aware of how similar it may be to previous functions, but it's interesting nonetheless. If you see any immediate problems with it, feel free to comment below. I can already imagine that you may have something to say. Hell, this is pretty much a function identical to what Norminals was meant to be. Now, consider the language M0. It is, in essence, first order logic. Now, M1 is in essence first order logic, but with an added symbol that allows it to predicate over the order of logic it is using. In essence, think of it as ath order logic, where a can be any variable defined within a previously established bth order logical expression. Let's use the symbol | for that. To clarify this, let's imagine the following, a pseudo-well-defined expression: "The largest number smaller than any finite number definable within |100|th order logic using 10^100 symbols or less.". Simple enough, right? Noticed how we called our language M1? Well, let's consider that "1" to be an order of its own. Now, let's let a new language, M1', use that "1" to describe orders. Notice what we have here? Well, let's skip a few steps and consider the following: Some other language, M2, can define orders using a system, within expressions themselves, that by listing collections of other symbols (or new symbols representing specific full expressions that use any other symbols), defines the order in question as the order predicating over all orders definable within the language defined by collection in question. The syntax for this would be something like Qa,b,c,d,(acb),... (Q being the order defined, a,b,c,... being the collection of symbols/expressions) Of course, there are many possibilities for ill-defined expressions in such a language (such as including all the symbols in M2 without some sort of limitation), but they don't interest us. Now, as you can see, we've got a new variable in question. That being, the a in Ma. How do we define that? Well, we know that we can do the same thing we did with the "1" from M1 in M2, just with the "2" in M2, to create M3. Let's call this a "meta-order" for now. Thus we can extent the system in question to any other "meta-order" we like. But wait, what if we allow a language to define the meta-order of logic within expressions of itself? Basically what we did with M2, but far more powerful. Let's call this language M0,1 for now. Now, I could go on to extent this array in many interesting ways, but you know what? I don't need to do that. Let the language M+ define the logic of an array-defined order (or meta-order or whatever). After all, it isn't too hard to define the array relations that would create M0,0,1, right? Now, for the function itself. Let M+(n) be the smallest number larger than any number defined by a well-defined expression in M+ using n symbols or less. Then, let M|0| be equal to 100, with M|n| being to M+(10^M|n-1|). The number? Let's call M|100| the Aldebaran, since it's the only I can think of now. Category:Blog posts